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Contract No. DE-AC36-08GO28308

Computation of Wave Loads 

under Multidirectional Sea 

States for Floating Offshore 

Wind Turbines

Preprint

T. Duarte

EDP Inovação

S. Gueydon

MARIN

J. Jonkman

National Renewable Energy Laboratory

A. Sarmento 

IST, University of Lisbon

To be presented at the 33rd International Conference on Ocean, 

Offshore and Arctic Engineering

San Francisco

June 8 – 13, 2014

Conference Paper

NREL/CP-5000-61161

March 2014
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Computation of Wave Loads under Multidirectional Sea States for 

Floating Offshore Wind Turbines

Tiago Duarte

EDP Inovação

Lisboa, Portugal

Sébastien Gueydon

MARIN

Netherlands

Jason Jonkman

National Renewable Energy Laboratory

Golden, Colorado, USA

António Sarmento

IST, University of Lisbon

Lisboa, Portugal

ABSTRACT

This paper focuses on the analysis of a floating wind 

turbine under multidirectional wave loading. Special attention 

is given to the different methods used to synthesize the 

multidirectional sea state. This analysis includes the doublesum and single-sum methods, as well as an equal-energy 

discretization of the directional spectrum. These three methods 

are compared in detail, including the ergodicity of the solution 

obtained. From the analysis, the equal-energy method proved to 

be the most computationally efficient while still retaining the 

ergodicity of the solution. This method was chosen to be 

implemented in the numerical code FAST. Preliminary results 

on the influence of these wave loads on a floating wind turbine 

showed significant additional roll and sway motion of the 

platform.

Keywords: Multidirectional Waves, Floating Wind Turbine, 

Synthesis of Multidirectional Seas. 

NOMENCLATURE

S(ω) frequency spectrum

D(θ) directional spectrum

N number of discrete frequencies

Θ number of discrete directions

ς(x, y, t) wave elevation at point (x, y) and time t

X(ω, θ) first-order load response amplitude operator

X+/− second-order load response amplitude operator 

(quadratic transfer function)

1. INTRODUCTION

Floating wind turbines are a promising technology to 

generate clean, renewable energy and are currently being 

studied with complex aero-hydro-servo-elastic modeling tools. 

Contributions from wave loads on the floating platform are 

most often modeled assuming unidirectional wave fields. 

However, the different wave components that comprise an 

irregular sea state come from different directions. Figure 1

compares the wave elevation obtained in the unidirectional case 

with a case involving multidirectional waves. 

Figure 1. Unidirectional (top) versus multidirectional 

sea surface (bottom), under a JONSWAP spectrum, 

with Hs=6 m, Tp=10 s, gamma = 2.2, and the 

spreading factor s=2. 

When modeling multidirectional sea states, the wave elevation 

is now a function not only of the wave frequency but also of the 

1

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incoming wave direction. In irregular waves, this is usually 

described with a directional spectrum or spreading function, 

similar to a frequency spectrum. It is assumed that the total 

spectrum can be defined as:

S(ω, θ) = S(ω).D(θ) (1)

where S(ω) is the frequency spectrum, independent of the 

direction of the waves, and D(θ) is the directional spectrum. 

There are several ways to define the directional spectrum. The 

most commonly used is a cosine spreading function:

D(θ) = C �cos �

π�θ−θ��

2θmax

��

2s

 , θ̅− θmmm < θ < θ̅+ θmmm (2)

where θ is the incoming wave direction, θ̅is the mean wave 

direction, θmmm is the maximum deviation from the mean wave 

direction, and s is the wave spreading parameter. C is a 

normalizing constant that is defined as [1]: 

C = √πΓ(s+1)

2θmaxΓ�s+1 2� �

 (3)

where Γ is the gamma function. This ensures that:

� D(θ)

θmax

−θmax

d = 1 (4)

This is a very important property that must be fulfilled by every 

method used to compute the discrete spectrum (see Section 

2.2). It guarantees that the total energy in each frequency band 

Δω is kept faithful to the frequency spectrum S(ω). 

The direction spectrum can also be a function of frequency, 

if the spreading coefficient is set to be frequency dependent. 

Despite being the most common directional spectrum, other 

methods have been developed as an alternative to the cos

2s

spectrum. Reference [2] describes several of these methods, 

which are also included in the MatLab toolbox WAFO [3]. 

2. SYNTHESIS OF MULTIDIRECTIONAL SEAS

The water surface elevation at a given point (x, y) on the 

still water plane at a given instant t as a result of a regular 

progressive wave can be described simply by: 

ς(x, y,t) = RR(A j�ωt−K(m.cos(θ)+y.sin(θ))�) (5)

where A is the wave amplitude (including phase) in meters, ω is 

the wave frequency in radians per second, K is the wave 

number in 1/m, and θ is the direction of the incoming wave. In 

an irregular sea state, the surface elevation can be described as 

the contribution from each individual wave:

ς(x, y,t) = RR(� �Ak,m j�ωkt−Kk(m.cos(θm)+y.sin(θm))�)

N

k=1

Θ

m=1

(6)

The indices m and k represent the wave direction and 

frequency component, respectively. The method of determining 

the individual wave components from a given wave spectrum is 

discussed in the following sections.

2.1. Frequency Spectrum Discretization

Several publications discuss the different approaches to 

discretize a wave spectrum (e.g., [4,5]). Depending on the way 

the frequency domain is discretized, the methods can be 

divided into constant frequency or a nonconstant frequency 

step. When using a constant frequency step, to obtain a unique 

wave elevation time series the number of individual frequency 

components is obtained from the desired length of the wave 

time-series. This is obtained from the sampling theorem:

N = 2π

∆t∆ω = tmmm∆t = 2ωmmm∆ω (7)

Despite the larger number of individual frequency components 

required, this approach allows the use of computationally

efficient fast Fourier transform (FFT) routines. This approach is 

implemented in FAST for unidirectional seas and was selected 

for this paper. 

A nonconstant frequency step discretization allows the 

number of frequency components to be reduced while still 

achieving a unique time-series. The reduced number of 

frequency components allows the summations in Eq. (6) to be 

directly computed. This approach is employed by the numerical 

code OrcaFlex [6] in combination with an equal-energy 

discretization routine.

2.2. Directional Spectrum Discretization

Similar to the frequency spectrum, the directional spectrum 

can also be subdivided into constant or nonconstant directional 

steps. When using constant direction steps, two methods of 

computing the wave elevation in directional sea states are found 

in the literature: the double-sum and single-sum methods [7]. 

A third method involving an equal-energy discretization of 

the directional spectrum is also used in this paper [6]. These are 

described in the following sections. Here, the wave amplitude is 

assumed to be deterministic with a random phase; the equations 

can be extended so that the amplitudes are also randomly 

distributed (an option available in FAST). 

2.2.1. Double-Sum Method

In the double-sum method, the wave spectrum is expressed 

as a two-dimensional matrix with size N by Θ, where N is the

number of frequency components and Θ is the number of 

directions considered. Therefore, this discretization contains 

several wave components with the same frequency over 

different directions. The wave elevation in this case is 

expressed as in Eq. (6) with the complex wave amplitude given 

by:

Ak,m = �2S(ωk)Δω.�D(θm)Δθ. j2πUk,m

(8)

where Uk,m represents a random number uniformly distributed 

between [0,1], obtained to give a random phase to each wave 

component with frequency ωk and direction θm. 

2.2.2. Single-Sum Method

The single-sum method avoids having different wave 

components with the same frequency by subdividing each wave 

frequency band into subfrequencies ∆ω′ = ∆ω⁄Θ. Therefore, 

the wave spectrum is expressed as a single vector of 

dimensions N × Θ, ensuring that the same wave frequency 

component does not repeat itself along the spectrum. 

References [8] and [1] suggest uniformly distributing the 

directions in ascending order across the subfrequencies. This 

should give the maximum frequency difference between waves 

2

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with the same direction. This method is presented in detail in 

[9]. The wave elevation is in this case expressed by:

ς(x, y,t) = RR(�Ak

j�ω′

kt−Kk(m.cos(θk)+y.sin(θk))�

)

N.Θ

k=1

(9)

where ω′

k is the subdivided frequency vector. The wave 

amplitude is obtained using:

Ak = �2S(ωk)Δω′. �D(θk)Δθ. j2πUk (10)

The distribution of wave directions θk is formally given by: 

θk = θmin + Δθ ′ with k′ = k k (Θ) (11)

where m (Θ) represents the arithmetic modulo Θ. This 

notation is often used in modular arithmetic function (or clock 

arithmetic) [10]. 

The time series obtained with this method will have N × Θ

points instead of just N points. However, only the first N points 

are used in the time-domain analysis (because tmmm

′ = tmmmΘ).

2.2.3. Equal-Energy Method

The last approach proposed in this paper is to discretize the 

directional spectrum using an equal-energy method. This is 

used in the commercial code OrcaFlex [6]. The idea behind this 

method is to use the same N points used in the unidirectional 

case. The equal-energy approach means that each direction

component represents the same energy. The discrete amplitude 

Ak is kept constant; therefore, the spectrum shape is achieved 

by using a nonconstant ΔΘ. The equal-energy discretization 

guarantees that there will be more direction components close 

to the mean wave direction. 

The method to determine the distribution of the discrete 

direction values can be summarized as:

1. Compute the spreading function D(θ) for a high number of 

equally spaced directions θ between −θmmm < θ < θmmm

(blue line in Figure 2). 

2. Compute the cumulative energy distribution P(θ) =

∫ D(θ′)dθ′ θ

−θmax (green line on Figure 2)[5]. 

3. Calculate the energy step vector, based on the user-defined 

number of directions Θ from 1

2Θ � to 1 − 12Θ � (the 

midpoints of each bin), with a step of 1

Θ� (circles on 

Figure 2). 

4. Interpolate the vector P(θ) to get the desired energy steps 

and determine the discrete values of θ′ (crosses on Figure 

2).

This approach should guarantee that the energy content in the 

frequency spectrum, discretized by a constant frequency step, is 

kept constant (eq. (4)) when using multidirectional waves. The 

wave elevation is given by:

ς(x, y,t) = RR(�Ak j�ωkt−Kk(m.cos(θk)+y.sin(θk))�)

N

k=1

(12)

where the wave amplitude is simply given by:

Ak = �2S(ωk)Δω. j2πUk (13)

For each frequency sample, one of the discrete directions 

obtained with the equal-energy approach should be randomly 

assigned. However, it should be ensured that each discrete 

direction should be used N/Θ times over the frequency 

spectrum. Therefore, it has to be ensured that N/Θ is an integer.

Figure 2. Spreading function, cumulative energy 

function, and discrete points (Θ = 5).

3. COMPARISON OF THE DIFFERENT 

DISCRETIZATION METHODS

To compare the three methods, the wave elevation and 

first-order loads were computed for a JONSWAP spectrum with 

Hs = 6 m and Tp = 10 s and gamma = 2.2. The directional 

spectrum was obtained with Eq. (2), with a spreading 

coefficient of s = 1 and evaluated between -90 and 90 degrees. 

For the double- and single-sum methods, 40 discrete directions 

were selected to suitably cover the wave direction spectrum, as 

recommended in [7]. However, for the equal-energy method, 

only 10 directions were necessary. This is further discussed in 

Section 3.4. The wave amplitude was obtained using a 

deterministic amplitude with a random phase. The 

unidirectional case is included for comparison purposes. 

Figure 3. Wave elevation using the different methods.

3.1. Wave-Elevation Time Series

The wave elevation at the origin was obtained with all 

three methods. Figure 3 presents the wave elevation time series 

at the origin (x, y) = (0,0). Because a random phase was used 

for each wave component, the time series do not exactly match 

-90 -75 -60 -45 -30 -15 0 15 30 45 60 75 90 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Direction θ [deg]

Spreading function

Cummulative Energy Distribution

Discrete Directions

Center of the Equal Energy bins

0 500 1000 1500 2000 2500 3000 3500 -5

-4

-3

-2

-1

0

1

2

3

4

5

Time [s]

Wave Elevation [m]

Unidirectional

Double-Sum

Single-Sum

Equal Energy

3

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between any of the methods except for the equal-energy 

method. In this case, the time series exactly matches the 

unidirectional case, as the phase was kept the same and only the 

wave direction of each frequency component was changed. 

The wave spectra can be seen in Figure 4, obtained by 

performing an FFT routine on the wave-elevation time-series 

using N points. The unidirectional case and the equal-energy 

method provide a smooth line as a result of the constant 

amplitude, random-phase spectrum used. The double-sum 

method presents a random amplitude spectrum because of the 

interaction of the different wave components with the same 

frequency and different directions and random phases. The 

single-sum method, despite having just one direction per 

frequency component, also produces this random amplitude 

spectrum, as only the first N points are used from the N × Θ

long time-series. The interaction between the different wave 

components is therefore artificially introduced.

Figure 4. Wave elevation spectra obtained with the 

different methods.

3.2. First-Order Loads

The first-order wave-excitation loads were calculated using 

the three presented methods. Table 1 presents the equations

used to compute the wave loads for the different methods 

previously described, assuming the reference point of the vessel 

has coordinates (x, y) = (0,0). The linear transfer function 

used is representative of the semisubmersible platform studied 

within the International Energy Agency (IEA) Wind Task 30 

Offshore Code Comparison Collaboration Continuation (OC4) 

project [11]. 

Table 1. Computation of first-order loads.

Method First-Order Loads

Double-Sum F(1)

(t) = RR(� �Ak,mX(ωk, θm) j(ωkt)

N

k=1

Θ

m=1

)

Single-Sum F(1)

(t) = RR(�AkX(ω′k, θk) j(ω′kt)

)

NΘ

k=1

Equal-Energy F(1)

(t) = RR(�AkX(ωk, θk) j(ωkt)

N

k=1

)

Figure 5 presents the time series for the force calculation in 

surge and sway. As a result of the random phase discussed 

earlier, none of the time series exactly match. For the 

unidirectional case, the sway forces are zero as expected, as the 

mean wave direction was assumed to be zero degrees. 

Figure 5. First-order excitation forces in surge (top) 

and sway (bottom) for the different methods.

Figure 6 presents the spectrum of the force time series. The 

single-sum and double-sum methods present a random 

amplitude spectrum because of the obtained random amplitude 

wave spectrum. The equal-energy method presents a much 

smoother spectrum. As it only considers a discrete number of 

directions, and most of them are close to the mean wave 

direction, all of the directional components have similar 

contributions to the force in the surge degree of freedom. The 

sway is smaller than the surge force as expected. The obtained 

spectrum for the equal-energy method now also presents 

random amplitude, according to the random direction assigned.

Figure 6. Spectra of the first-order excitation forces 

for surge (top) and sway (bottom).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0

1000

2000

3000

4000

5000

6000

Frequency [rad/s]

Wave Elevation

Unidirectional

Double-Sum

Single-Sum

Equal Energy

0 100 200 300 400 500 600 -2

-1

0

1

2 x 107

Time [s]

FSurge [N]

Unidirectional

Double-Sum

Single-Sum

Equal Energy

0 100 200 300 400 500 600 -2

-1

0

1

2

x 107

Time [s]

FSway [N]

Unidirectional

Double-Sum

Single-Sum

Equal Energy

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.5

1

1.5

2

x 1010

Frequency [rad/s]

F Surge

Unidirectional

Double-Sum

Single-Sum

Equal Energy

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0

5

10

x 109

Frequency [rad/s]

F Sway

Unidirectional

Double-Sum

Single-Sum

Equal Energy

4

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Figure 7 presents the ratio of computational time for the 

different methods compared with the unidirectional case, for a 

3,600-s simulation. The equal-energy method used 10 discrete 

directions, whereas the other methods used 40. The single-sum 

method presents the highest total value (around 550 times 

higher than the unidirectional case), as a result of the 

interpolation of the frequency spectrum, which generates a 

vector with approximately 1.44 million entries. The equalenergy method is faster than the double-sum method. This 

difference should be much larger in the computation of the 

second-order forces.

Figure 7. Computational time required for the 

different methods.

3.3. Second-Order Loads

Second-order wave loads require the computation of 

quadratic transfer functions, both for the difference- and the 

sum-frequency problems (respectively represented by X− and 

X+). For multidirectional sea states, these matrices are fourdimensional, taking into account the second-order interactions 

between waves with different frequencies and directions. 

Table 2. Computation of second-order loads.

Method Second-Order Loads

Double-Sum

F(2)(t)

= RR �� � ���Ak,mAl,nX+(ωk, ωl, θm, θn) j(ωk+ωl)t

N

l=1

N

k=1

Θ

m=1

Θ

c=1

+ Ak,mAl,n

∗

X−(ωk, ωl, θm, θn) j(ωk−ωl)t��

Single-Sum

F(2)(t)

= RR ����AkAlX+(ω′k, ω′l, θk, θl) j(ω′k+ω′l)t

NΘ

l=1

NΘ

k=1

+ AkAl

∗

X−(ω′k, ω′l, θk, θl) j(ω′k−ω′l)t ��

Equal-Energy

F(2)(t)

= RR ����AkAlX+(ωk, ωl, θk,θl) j(ωk+ωl)t

N

l=1

N

k=1

+ AkAl

∗

X−(ωk, ωl, θk, θl) j(ωk−ωl)t��

These transfer functions can be obtained in the frequency 

domain with a second-order panel code, like WAMIT [12]. 

Table 2 presents the required equations to compute these loads. 

As can be seen, the double sum-method now requires a 

quadruple sum over two directions and two frequencies. The 

efficiency of the equal-energy approach is very valuable when 

studying these loads. For this paper, it was not possible to 

obtain the required transfer functions, and therefore the analysis 

of these loads will be included in further study. The importance 

of the second-order loads under multidirectional sea states has 

been highlighted in [13]. 

3.4. Ergodicity

Ocean waves are usually assumed to be a random ergodic 

process (stationary and homogenous). The spatial variability is 

the phenomenon most affected by the different methods used to 

synthesize multidirectional seas. It has been discussed in the 

literature how the double-sum method affects the mean wave 

energy across the domain because of the interaction of waves 

with the same frequency and different directions [14]. The other 

important quantity is the variability of the cross-spectrum 

between two points in space. This quantity is used to determine 

bidirectional spectra in measurements both in the tank and in 

the ocean. 

3.4.1. Mean Energy

The ergodicity assumption requires that the mean energy 

across the sea surface is constant. In real sea conditions, this is 

only untrue in the presence of significant reflected waves, 

mainly close to shore. In a wave tank, significant reflection can 

occur depending on the tank configuration and the synthesized 

waves. However, the energy content should still be constant 

along the tank. The mean energy can obtained integrating the 

energy spectrum for each location [15]: 

E� = ρρ � � S(ω, θ)d d

2π

0

∞

0

(14)

Or in the discrete form:

E� = 1

2 ρρ|Z(ωk)| ����������2� (15)

where Z(ωk) = Ak

N

2� . To compare the discretization 

methods, the mean wave energy was computed for a square 

grid of 1 km by 1 km, in intervals of 10 m. The singlesummation and equal-energy methods provided constant energy 

content across the domain. However, for the double-summation 

method, we obtained the pattern seen in Figure 8. This result is 

consistent with the one found in [14]. 

Figure 8. Mean wave energy across the domain for 

the double-sum method. 

Double-Sum Single-Sum Equal energy 0

200

400

600

800

1000

% of the Uni-Directional Case

5523%

5

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As stated in [1], the only way to reduce this variability is to 

increase the length of the simulation or by averaging different 

realizations. Figure 9 presents the normalized wave energy for 

different simulation lengths. The energy content varies within a 

5% range for all the realizations, and it is only reduced to 1% 

for simulations with 9 hours. The variability is not that high 

because the mean energy is computed for the entire frequency 

range. Reference [1] presents results for a smaller frequency 

band, where a variability of up to 60% along the domain was 

encountered for the shorter simulations. 

Figure 9. Normalized mean wave energy for the 

double-sum method and different lengths of the 

simulation.

One parameter that needs to be taken into account for the 

single-summation method is the number of directions 

considered. This value needs to ensure a good discretization, 

especially close to the peak region of the directional spectrum 

(close to the mean wave direction). The directional step should 

be sufficiently small to guarantee that the energy content of the 

mean wave direction is captured, and this is especially critical 

for peaked spectrum (high values of the spreading coefficient). 

Figure 10 shows the mean wave energy for the single-sum 

method and the equal-energy method. The single-sum method 

seems to have converged for a number of directions higher than 

15. It should be noted that [1] and [14] suggest that a minimum 

number of 32 directions should be used to guarantee the 

accuracy of this method, based on the cross-spectrum analysis. 

The equal-energy discretization guarantees a constant mean 

wave energy along the domain even when using just two 

discrete directions.

Figure 10. Mean wave energy for the single-sum and 

the equal-energy method.

3.4.2. Cross-Spectrum

The cross-spectrum between the wave elevations at 

different locations is used to measure wave directionality in 

wave basins. This section presents the variability of the crossspectrum between two points. The cross-spectrum between two 

points p and q is given by [4]: 

Spq(ω) = Zp

∗ (ω)Zq(ω)∆t

π (16)

Across the domain, the ergodicity assumption requires that the 

variability is constant along the domain:

VVV�Spq(ω)� = VVV �

Zp

∗ (ω)Zq(ω)∆t

π � = c c . (17)

Figure 11 presents the variance of the cross-spectrum relative to 

the origin of the referential for the double-sum method. As 

shown there is a significant variance along the domain, despite 

using a 3-hr simulation time. 

Figure 11. Variance of the cross-spectrum obtained 

with the double-sum method.

Figure 12 shows the same result for the single-sum and equalenergy method. Despite not being exactly constant across the 

domain because of numerical issues, the difference across the 

domain is smaller than 1%.

(A) (B)

Figure 12. Variance of the cross-spectrum obtained 

with the single-sum method (A) and equal-energy 

method (B).

These results are compared along the line y = 0, in Figure 13, 

normalized by the mean value of the variance cross-spectrum.

Once again it can be seen that the double-sum method does not 

provide a homogeneous and ergodic sea state.

-1000 -800 -600 -400 -200 0 200 400 600 800 1000 0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

x [m]

Normalized Mean Wave Energy

 

t

max=10 min

t

max=30 min

t

max=1 hour

t

max=3 hours

t

max=9 hours

0 5 10 15 20 25 30 35 40 1.12

1.125

1.13

1.135

1.14 x 104

Number of Directions

Mean Wave Energy

Single-Sum Method

Equal Energy Method

-1000

-500

0

500

1000

-1000

-500

0

500

1000

0

1

2

3

4

5

x 104

(yp-y0 (x ) [m] p-x0) [m]

Variance

-1000

-500

0

500

1000

-1000

-500

0

500

1000

3.745

3.75

3.755

3.76

3.765

3.77

3.775

x 106

(yp-y0) [m] (xp-x0) [m]

Variance

-1000

-500

0

500

1000

-1000

-500

0

500

1000

4.66

4.68

4.7

4.72

4.74

x 104

(yp

-y0) [m] (xp-x0

) [m]

Variance

6

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Figure 13. Variance of the cross-spectrum along y = 0 

for the different methods.

4. PRELIMINARY ANALYSIS OF A FLOATING 

OFFSHORE WIND TURBINE UNDER 

MULTIDIRECTIONAL SEAS 

The OC4 semisubmersible platform with the National 

Renewable Energy Laboratory’s 5-MW reference offshore wind 

turbine was considered for this study [11]. The platform 

consists of three columns plus a central column where the 

tower is located, as seen in Figure 14. 

Figure 14. OC4 semisubmersible platform.

The equal-energy method was selected for this analysis. The 

first-order wave-excitation loads were calculated in MatLab 

and fed to FAST, which performed the time-marching 

simulation. The load case chosen corresponds to the JONSWAP 

spectrum previously mentioned. A spreading coefficient of one

was selected for the multidirectional case. No wind loads were 

considered in this analysis.

Figure 15 presents the platform motions for this case, with 

a unidirectional sea state and a multidirectional case. For both 

cases, the main wave direction is aligned with the x-axis of the 

platform, which represents the surge motion. 

As expected, the platform has basically no sway and roll 

motion in the unidirectional sea. However, for the 

multidirectional case, there is a significant additional motion of 

the platform in these degrees of freedom. The sway and roll 

motion found represents roughly 20% of the surge and pitch 

motion, respectively. 

Figure 15. Platform motions under irregular sea state, 

with and without multidirectional loading.

5. CONCLUSIONS

In this paper we reviewed different methods to synthesize 

the wave elevation under multidirectional sea states. The 

double-sum method is the most commonly mentioned method

in the literature, but does not provide an ergodic solution. The 

single-sum method fixed this problem, by shifting the different 

frequency components and avoiding the interaction between 

them. However, it still requires significant computational time, 

especially for long simulations. The equal-energy method 

described here combines a constant frequency-step 

-1000 -800 -600 -400 -200 0 200 400 600 800 1000 0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

(xp-x0) [m]

Normalized Variance of the Cross-Spectrum

Double-Sum

Single-Sum

Equal energy

1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 -2

-1

0

1

2

Time (s)

 Surge (m)

Uni-Directional

Multi-Directional

1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 -1

-0.5

0

0.5

1

Time (s)

 Sway (m)

Uni-Directional

Multi-Directional

1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 -2

-1

0

1

2

Time (s)

 Pitch (o)

Uni-Directional

Multi-Directional

1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 -1

-0.5

0

0.5

1

Time (s)

 Roll (o)

Uni-Directional

Multi-Directional

7

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.
discretization with an equal-energy discretization of the 

directional spectrum. This allows using the same number of 

frequency components for both the unidirectional and 

multidirectional cases, thus saving significant computational 

effort. This method also proved to ensure the ergodicity 

proprieties of the wave spectrum. Based on these results, this 

method was selected to be implemented within FAST and 

should become available in a future release.

In addition, a preliminary study was performed on the OC4 

semisubmersible platform. The comparison between the 

unidirectional and multidirectional sea state without wind loads 

showed a significant increase in the platform sway and roll 

motion. These findings should motivate further studies to 

carefully assess the impact of the multidirectional loads on the 

platform’s ultimate loads and fatigue life.

6. ACKNOWLEDGMENTS 

This work was supported by the U.S. Department of 

Energy under Contract No. DE-AC36-08-GO28308 with the 

National Renewable Energy Laboratory.

7. REFERENCES

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[4] Kim, C. H., 2008, “Nonlinear Waves and offshore 

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